Welcome to AP Calculus.
Your journey into calculus starts with the topic of limits. Why? Because limits make calculus work. The two big things in calculus are called the derivative and the definite integral; both are limits.
The first use of limits will be when you study continuity. A continuous function is one, roughly speaking, whose graph can be drawn without taking your pen off the paper. Limits will make this concept firm mathematically.
After that, if you flip through your book, you won’t see that many limits after the limit chapter. You will see derivatives and definite integrals – they are limits under a different name.
Notation: There is a new notation to learn for limits. It looks like this 
You will start by finding the limits of various functions. Sometimes the limit is easy to find – just substitute the number x is approaching into the (that’s what happened above); what you get is the limit. If a limit exists, it is a number. If you get a number by substituting, usually that’s the limit – end of problem.
But other times you get expressions that are not numbers when you substitute (like maybe you end up trying to divide by zero). In that case, you will need to do some sort of algebraic or trigonometric simplification. Your teacher will help you learn the “tricks” involved. Derivatives and the definite integral are both limits involving dividing by zero.
Some limits may not exist at all; in this case, you say, “Does not exist” or “DNE.” Others do not exist, but we say they are “equal to infinity.” Infinity will be the subject of the next post in this series.
As you learn to find limits, look for patterns. The limits of similar looking expressions are often found in similar ways.
One good way to see what a limit is, or is not, is to graph the expression. Use your graphing calculator.
Your calculator may “misinform” you sometimes! But even that is a help. (Hint: when your calculator does misinform you, about limits or anything else, that’s a time to look deeper into the situation: something interesting is going on.)
Producing a table of values (on your calculator) can often help you see what’s happening, as well. (Hint: while tables are useful, what happens between the values in the table is not always clear; that’s where the trouble may be.)
The first thing you will use limits for is to investigate continuity. When limits do not exist, continuity is usually the problem. Continuity will be the subject of a later post.
So, get ready for your trip through calculus: it’s an unlimited journey.
The next post in this series, “Why Infinity?”, will appear on Friday August 25, 2023.
Course and Exam Description: Unit 1 all topics
Teaching Calculus 